Mixed logical dynamical (MLD) systems, as introduced e.g. in the article “Control of Systems Integrating Logic, Dynamics, and Constraints”, A. Bemporad and M. Morari, Automatica 35(3), 1999, pp 407-427, represent a mathematical framework for modeling systems described by interacting physical laws, logical rules, and operating constraints, generally called “hybrid systems”. MLD systems are determined or described by linear dynamic equations subject to linear mixed-integer inequalities involving both continuous, i.e. real-valued, and binary, i.e. boolean-valued, variables. The variables include continuous and binary states x, inputs u and outputs y, as well as continuous auxiliary variables z and binary auxiliary variables δ, as described in the following equations:x(t+1)=Ax(t)+B1 u(t)+B2δ(t)+B3z(t)  (Eq. 1a)y(t)=Cx(t)+D1u(t)+D2δ(t)+D3z(t)  (Eq. 1b)E2δ(t)+E3z(t)<=E1u(t)+E4x(t)+E5  (Eq. 1c)In general, the variables mentioned are vectors and A, Bi, C, Di, Ei are matrices of suitable dimensions.
In order to be well posed, the MLD system (Eqs. 1) must be such that for any given x(t) and u(t) the values of δ(t) and z(t) are defined uniquely. Formulations or relationships in the form of Eqs. 1 appear naturally when logical statements are written as propositional calculus expressions, or when bounds on the states are set explicitly. Among the advantages of the MLD framework are the possibility to automatically generate the matrices of MLD systems from a high-level description. MLD systems generalize a wide set of models, among which there are linear hybrid systems and even nonlinear systems whose nonlinearities can be expressed or at least suitably approximated by piecewise linear functions.
The drawback of the MLD systems approach is the relative complexity of the theory, which in turn makes the modelling and maintenance of a complex industrial system difficult for people without a background in mixed integer optimization.
Model predictive control (MPC) is a procedure of solving an optimal-control problem, which includes system dynamics and constraints on the system output, state and input variables. The main idea of model predictive control is to use a model of the plant or process, valid at least around a certain operating point, to predict the future evolution of the system. Based on this prediction, at each time step t the controller selects a sequence of future command inputs or control signals through an on-line optimization procedure, which aims at optimizing a performance, cost or objective function, and enforces fulfillment of the constraints. Only the first sample of the optimal sequence of future command inputs is actually applied to the system at time t. At time t+1, a new sequence is evaluated to replace the previous one. This on-line re-planning provides the desired feedback control feature.
Model Predictive Control can be applied for stabilizing a MLD system to an equilibrium state or to track a desired reference trajectory via feedback control. Standard routines can then be used to recast the previous control problem into a Mixed Integer Linear Programming (MILP) of the following form:
                                                        min                              v                _                                      ⁢                          f              ⁡                              (                                  v                  _                                )                                              =                                                    min                                  v                  _                                            ⁢                                                                    g                    _                                    T                                ⁢                                  v                  _                                ⁢                                                                  ⁢                                  s                  .                  t                  .                                                                          ⁢                  A                                ⁢                                  v                  _                                                      ≤                          b              _                                      ,                            (                  Eq          ⁢          .2                )            where the optimization vector v includes the aforementioned MLD input and auxiliary variables u, δ, z via vT=[uT δT zT], the vector g denotes a cost vector, the matrix A is a constraint matrix and the vector b is termed a boundary vector. Standard procedures can be adapted to handle time varying parameters such as cost and price coefficients. The solution of the MILP problem can be obtained by reverting to the commercial optimisation problem solver for solving linear, mixed-integer and quadratic programming problems called CPLEX (http://www.ilog.com/products/cplex/).
Model predictive control (MPC) in combination with Mixed Logical Dynamical (MLD) systems descriptions has been used for modeling and control of processes in the utility automation and process industry. By way of example, a method of scheduling a cement production is described in the article “Using Model Predictive Control and Hybrid Systems for Optimal Scheduling of Industrial Processes”, by E. Gallestey et al., AT Automatisierungstechnik, Vol. 51, no. 6, 2003, pp. 285-293.